LMFDB - Embedded newform 1045.4.a.c.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,4,Mod(1,1045)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1045.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1045.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$61.6569959560$
Analytic rank:	$1$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{19} - 105 x^{18} + 103 x^{17} + 4500 x^{16} - 4345 x^{15} - 101844 x^{14} + 95592 x^{13} + \cdots + 150528 $ x^20 - x^19 - 105x^18 + 103x^17 + 4500x^16 - 4345x^15 - 101844x^14 + 95592x^13 + 1317797x^12 - 1160501x^11 - 9914845x^10 + 7570653x^9 + 42786958x^8 - 23777633x^7 - 102801526x^6 + 28436356x^5 + 122325928x^4 + 411232x^3 - 47350496x^2 - 4782848x + 150528
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.35645$ of defining polynomial
Character	$\chi$	$=$	1045.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.35645 q^{2} -6.45217 q^{3} -2.44713 q^{4} -5.00000 q^{5} +15.2042 q^{6} +27.1536 q^{7} +24.6182 q^{8} +14.6305 q^{9} +O(q^{10})$	\(q-2.35645 q^{2} -6.45217 q^{3} -2.44713 q^{4} -5.00000 q^{5} +15.2042 q^{6} +27.1536 q^{7} +24.6182 q^{8} +14.6305 q^{9} +11.7823 q^{10} -11.0000 q^{11} +15.7893 q^{12} -62.2421 q^{13} -63.9861 q^{14} +32.2608 q^{15} -38.4345 q^{16} -24.6401 q^{17} -34.4760 q^{18} +19.0000 q^{19} +12.2357 q^{20} -175.199 q^{21} +25.9210 q^{22} -87.4691 q^{23} -158.841 q^{24} +25.0000 q^{25} +146.670 q^{26} +79.8102 q^{27} -66.4484 q^{28} +221.769 q^{29} -76.0211 q^{30} -214.126 q^{31} -106.376 q^{32} +70.9739 q^{33} +58.0631 q^{34} -135.768 q^{35} -35.8027 q^{36} -192.330 q^{37} -44.7726 q^{38} +401.596 q^{39} -123.091 q^{40} +323.178 q^{41} +412.849 q^{42} +382.510 q^{43} +26.9185 q^{44} -73.1524 q^{45} +206.117 q^{46} -12.8708 q^{47} +247.986 q^{48} +394.316 q^{49} -58.9113 q^{50} +158.982 q^{51} +152.315 q^{52} -345.779 q^{53} -188.069 q^{54} +55.0000 q^{55} +668.471 q^{56} -122.591 q^{57} -522.587 q^{58} +318.805 q^{59} -78.9466 q^{60} -461.116 q^{61} +504.577 q^{62} +397.270 q^{63} +558.147 q^{64} +311.210 q^{65} -167.246 q^{66} +868.796 q^{67} +60.2975 q^{68} +564.365 q^{69} +319.930 q^{70} +617.212 q^{71} +360.176 q^{72} +1124.99 q^{73} +453.217 q^{74} -161.304 q^{75} -46.4955 q^{76} -298.689 q^{77} -946.342 q^{78} +611.159 q^{79} +192.172 q^{80} -909.972 q^{81} -761.552 q^{82} +302.346 q^{83} +428.736 q^{84} +123.200 q^{85} -901.366 q^{86} -1430.89 q^{87} -270.800 q^{88} -1000.18 q^{89} +172.380 q^{90} -1690.09 q^{91} +214.049 q^{92} +1381.57 q^{93} +30.3293 q^{94} -95.0000 q^{95} +686.358 q^{96} -599.833 q^{97} -929.188 q^{98} -160.935 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9}+O(q^{10}) $ 20 * q - q^2 - 8 * q^3 + 51 * q^4 - 100 * q^5 - 54 * q^6 + 49 * q^7 + 9 * q^8 + 146 * q^9	\( 20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9} + 5 q^{10} - 220 q^{11} - 59 q^{12} + 60 q^{13} - 89 q^{14} + 40 q^{15} + 275 q^{16} - 155 q^{17} + 45 q^{18} + 380 q^{19} - 255 q^{20} + 105 q^{21} + 11 q^{22} - 154 q^{23} - 397 q^{24} + 500 q^{25} + 176 q^{26} - 206 q^{27} + 155 q^{28} - 305 q^{29} + 270 q^{30} - 759 q^{31} - 254 q^{32} + 88 q^{33} - 565 q^{34} - 245 q^{35} + 705 q^{36} + 698 q^{37} - 19 q^{38} - 758 q^{39} - 45 q^{40} + 547 q^{41} + 106 q^{42} - 925 q^{43} - 561 q^{44} - 730 q^{45} - 254 q^{46} - 681 q^{47} - 540 q^{48} + 213 q^{49} - 25 q^{50} - 899 q^{51} + 889 q^{52} - 419 q^{53} - 2241 q^{54} + 1100 q^{55} - 2473 q^{56} - 152 q^{57} - 1440 q^{58} - 2829 q^{59} + 295 q^{60} - 959 q^{61} + 1575 q^{62} - 426 q^{63} + 93 q^{64} - 300 q^{65} + 594 q^{66} - 1020 q^{67} - 4218 q^{68} - 572 q^{69} + 445 q^{70} + 106 q^{71} + 210 q^{72} + 558 q^{73} - 3439 q^{74} - 200 q^{75} + 969 q^{76} - 539 q^{77} - 3599 q^{78} + 536 q^{79} - 1375 q^{80} - 2128 q^{81} - 1255 q^{82} - 4179 q^{83} - 2024 q^{84} + 775 q^{85} - 1119 q^{86} - 557 q^{87} - 99 q^{88} - 4120 q^{89} - 225 q^{90} - 111 q^{91} - 2831 q^{92} + 801 q^{93} + 1213 q^{94} - 1900 q^{95} - 6147 q^{96} + 1414 q^{97} - 7869 q^{98} - 1606 q^{99}+O(q^{100}) \) 20 * q - q^2 - 8 * q^3 + 51 * q^4 - 100 * q^5 - 54 * q^6 + 49 * q^7 + 9 * q^8 + 146 * q^9 + 5 * q^10 - 220 * q^11 - 59 * q^12 + 60 * q^13 - 89 * q^14 + 40 * q^15 + 275 * q^16 - 155 * q^17 + 45 * q^18 + 380 * q^19 - 255 * q^20 + 105 * q^21 + 11 * q^22 - 154 * q^23 - 397 * q^24 + 500 * q^25 + 176 * q^26 - 206 * q^27 + 155 * q^28 - 305 * q^29 + 270 * q^30 - 759 * q^31 - 254 * q^32 + 88 * q^33 - 565 * q^34 - 245 * q^35 + 705 * q^36 + 698 * q^37 - 19 * q^38 - 758 * q^39 - 45 * q^40 + 547 * q^41 + 106 * q^42 - 925 * q^43 - 561 * q^44 - 730 * q^45 - 254 * q^46 - 681 * q^47 - 540 * q^48 + 213 * q^49 - 25 * q^50 - 899 * q^51 + 889 * q^52 - 419 * q^53 - 2241 * q^54 + 1100 * q^55 - 2473 * q^56 - 152 * q^57 - 1440 * q^58 - 2829 * q^59 + 295 * q^60 - 959 * q^61 + 1575 * q^62 - 426 * q^63 + 93 * q^64 - 300 * q^65 + 594 * q^66 - 1020 * q^67 - 4218 * q^68 - 572 * q^69 + 445 * q^70 + 106 * q^71 + 210 * q^72 + 558 * q^73 - 3439 * q^74 - 200 * q^75 + 969 * q^76 - 539 * q^77 - 3599 * q^78 + 536 * q^79 - 1375 * q^80 - 2128 * q^81 - 1255 * q^82 - 4179 * q^83 - 2024 * q^84 + 775 * q^85 - 1119 * q^86 - 557 * q^87 - 99 * q^88 - 4120 * q^89 - 225 * q^90 - 111 * q^91 - 2831 * q^92 + 801 * q^93 + 1213 * q^94 - 1900 * q^95 - 6147 * q^96 + 1414 * q^97 - 7869 * q^98 - 1606 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.35645	−0.833132	−0.416566	−	0.909106i	$-0.636766\pi$
$2$	−2.35645	−0.833132	−0.416566	+	0.909106i	$0.636766\pi$
$3$	−6.45217	−1.24172	−0.620860	−	0.783921i	$-0.713217\pi$
$3$	−6.45217	−1.24172	−0.620860	+	0.783921i	$0.713217\pi$
$4$	−2.44713	−0.305892
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	15.2042	1.03452
$7$	27.1536	1.46616	0.733078	−	0.680145i	$-0.238083\pi$
$7$	27.1536	1.46616	0.733078	+	0.680145i	$0.238083\pi$
$8$	24.6182	1.08798
$9$	14.6305	0.541870
$10$	11.7823	0.372588
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	15.7893	0.379832
$13$	−62.2421	−1.32791	−0.663955	−	0.747772i	$-0.731124\pi$
$13$	−62.2421	−1.32791	−0.663955	+	0.747772i	$0.731124\pi$
$14$	−63.9861	−1.22150
$15$	32.2608	0.555314
$16$	−38.4345	−0.600539
$17$	−24.6401	−0.351535	−0.175767	−	0.984432i	$-0.556241\pi$
$17$	−24.6401	−0.351535	−0.175767	+	0.984432i	$0.556241\pi$
$18$	−34.4760	−0.451449
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	12.2357	0.136799
$21$	−175.199	−1.82055
$22$	25.9210	0.251199
$23$	−87.4691	−0.792981	−0.396491	−	0.918039i	$-0.629772\pi$
$23$	−87.4691	−0.792981	−0.396491	+	0.918039i	$0.629772\pi$
$24$	−158.841	−1.35097
$25$	25.0000	0.200000
$26$	146.670	1.10632
$27$	79.8102	0.568870
$28$	−66.4484	−0.448485
$29$	221.769	1.42005	0.710024	−	0.704177i	$-0.248684\pi$
$29$	221.769	1.42005	0.710024	+	0.704177i	$0.248684\pi$
$30$	−76.0211	−0.462650
$31$	−214.126	−1.24058	−0.620292	−	0.784371i	$-0.712986\pi$
$31$	−214.126	−1.24058	−0.620292	+	0.784371i	$0.712986\pi$
$32$	−106.376	−0.587652
$33$	70.9739	0.374393
$34$	58.0631	0.292875
$35$	−135.768	−0.655684
$36$	−35.8027	−0.165753
$37$	−192.330	−0.854566	−0.427283	−	0.904118i	$-0.640529\pi$
$37$	−192.330	−0.854566	−0.427283	+	0.904118i	$0.640529\pi$
$38$	−44.7726	−0.191134
$39$	401.596	1.64889
$40$	−123.091	−0.486559
$41$	323.178	1.23102	0.615510	−	0.788129i	$-0.288950\pi$
$41$	323.178	1.23102	0.615510	+	0.788129i	$0.288950\pi$
$42$	412.849	1.51676
$43$	382.510	1.35656	0.678282	−	0.734802i	$-0.262725\pi$
$43$	382.510	1.35656	0.678282	+	0.734802i	$0.262725\pi$
$44$	26.9185	0.0922298
$45$	−73.1524	−0.242331
$46$	206.117	0.660658
$47$	−12.8708	−0.0399445	−0.0199723	−	0.999801i	$-0.506358\pi$
$47$	−12.8708	−0.0399445	−0.0199723	+	0.999801i	$0.506358\pi$
$48$	247.986	0.745701
$49$	394.316	1.14961
$50$	−58.9113	−0.166626
$51$	158.982	0.436508
$52$	152.315	0.406197
$53$	−345.779	−0.896159	−0.448080	−	0.893994i	$-0.647892\pi$
$53$	−345.779	−0.896159	−0.448080	+	0.893994i	$0.647892\pi$
$54$	−188.069	−0.473944
$55$	55.0000	0.134840
$56$	668.471	1.59515
$57$	−122.591	−0.284870
$58$	−522.587	−1.18309
$59$	318.805	0.703472	0.351736	−	0.936099i	$-0.385591\pi$
$59$	318.805	0.703472	0.351736	+	0.936099i	$0.385591\pi$
$60$	−78.9466	−0.169866
$61$	−461.116	−0.967866	−0.483933	−	0.875105i	$-0.660792\pi$
$61$	−461.116	−0.967866	−0.483933	+	0.875105i	$0.660792\pi$
$62$	504.577	1.03357
$63$	397.270	0.794465
$64$	558.147	1.09013
$65$	311.210	0.593860
$66$	−167.246	−0.311918
$67$	868.796	1.58418	0.792092	−	0.610401i	$-0.208992\pi$
$67$	868.796	1.58418	0.792092	+	0.610401i	$0.208992\pi$
$68$	60.2975	0.107532
$69$	564.365	0.984661
$70$	319.930	0.546271
$71$	617.212	1.03168	0.515842	−	0.856684i	$-0.327479\pi$
$71$	617.212	1.03168	0.515842	+	0.856684i	$0.327479\pi$
$72$	360.176	0.589543
$73$	1124.99	1.80371	0.901854	−	0.432041i	$-0.142206\pi$
$73$	1124.99	1.80371	0.901854	+	0.432041i	$0.142206\pi$
$74$	453.217	0.711966
$75$	−161.304	−0.248344
$76$	−46.4955	−0.0701764
$77$	−298.689	−0.442062
$78$	−946.342	−1.37375
$79$	611.159	0.870389	0.435194	−	0.900337i	$-0.356680\pi$
$79$	611.159	0.870389	0.435194	+	0.900337i	$0.356680\pi$
$80$	192.172	0.268569
$81$	−909.972	−1.24825
$82$	−761.552	−1.02560
$83$	302.346	0.399840	0.199920	−	0.979812i	$-0.435932\pi$
$83$	302.346	0.399840	0.199920	+	0.979812i	$0.435932\pi$
$84$	428.736	0.556893
$85$	123.200	0.157211
$86$	−901.366	−1.13020
$87$	−1430.89	−1.76330
$88$	−270.800	−0.328038
$89$	−1000.18	−1.19123	−0.595614	−	0.803271i	$-0.703091\pi$
$89$	−1000.18	−1.19123	−0.595614	+	0.803271i	$0.703091\pi$
$90$	172.380	0.201894
$91$	−1690.09	−1.94692
$92$	214.049	0.242566
$93$	1381.57	1.54046
$94$	30.3293	0.0332791
$95$	−95.0000	−0.102598
$96$	686.358	0.729699
$97$	−599.833	−0.627874	−0.313937	−	0.949444i	$-0.601648\pi$
$97$	−599.833	−0.627874	−0.313937	+	0.949444i	$0.601648\pi$
$98$	−929.188	−0.957777
$99$	−160.935	−0.163380
$100$	−61.1783	−0.0611783
$101$	−1597.42	−1.57375	−0.786876	−	0.617111i	$-0.788303\pi$
$101$	−1597.42	−1.57375	−0.786876	+	0.617111i	$0.788303\pi$
$102$	−374.633	−0.363669
$103$	647.478	0.619397	0.309698	−	0.950835i	$-0.399772\pi$
$103$	647.478	0.619397	0.309698	+	0.950835i	$0.399772\pi$
$104$	−1532.29	−1.44474
$105$	875.997	0.814177
$106$	814.812	0.746619
$107$	284.071	0.256656	0.128328	−	0.991732i	$-0.459039\pi$
$107$	284.071	0.256656	0.128328	+	0.991732i	$0.459039\pi$
$108$	−195.306	−0.174013
$109$	−422.288	−0.371081	−0.185541	−	0.982637i	$-0.559404\pi$
$109$	−422.288	−0.371081	−0.185541	+	0.982637i	$0.559404\pi$
$110$	−129.605	−0.112339
$111$	1240.95	1.06113
$112$	−1043.63	−0.880483
$113$	776.152	0.646144	0.323072	−	0.946374i	$-0.395284\pi$
$113$	776.152	0.646144	0.323072	+	0.946374i	$0.395284\pi$
$114$	288.880	0.237334
$115$	437.345	0.354632
$116$	−542.697	−0.434381
$117$	−910.631	−0.719554
$118$	−751.249	−0.586085
$119$	−669.066	−0.515404
$120$	794.203	0.604171
$121$	121.000	0.0909091
$122$	1086.60	0.806360
$123$	−2085.20	−1.52858
$124$	523.994	0.379484
$125$	−125.000	−0.0894427
$126$	−936.147	−0.661894
$127$	−1294.76	−0.904659	−0.452330	−	0.891851i	$-0.649407\pi$
$127$	−1294.76	−0.904659	−0.452330	+	0.891851i	$0.649407\pi$
$128$	−464.235	−0.320570
$129$	−2468.02	−1.68447
$130$	−733.352	−0.494763
$131$	−1829.24	−1.22001	−0.610005	−	0.792398i	$-0.708832\pi$
$131$	−1829.24	−1.22001	−0.610005	+	0.792398i	$0.708832\pi$
$132$	−173.682	−0.114524
$133$	515.918	0.336359
$134$	−2047.28	−1.31983
$135$	−399.051	−0.254406
$136$	−606.593	−0.382463
$137$	934.479	0.582758	0.291379	−	0.956608i	$-0.405886\pi$
$137$	934.479	0.582758	0.291379	+	0.956608i	$0.405886\pi$
$138$	−1329.90	−0.820352
$139$	−2010.79	−1.22700	−0.613501	−	0.789694i	$-0.710239\pi$
$139$	−2010.79	−1.22700	−0.613501	+	0.789694i	$0.710239\pi$
$140$	332.242	0.200568
$141$	83.0443	0.0496000
$142$	−1454.43	−0.859528
$143$	684.663	0.400380
$144$	−562.315	−0.325414
$145$	−1108.84	−0.635065
$146$	−2651.00	−1.50273
$147$	−2544.20	−1.42750
$148$	470.658	0.261405
$149$	−1257.27	−0.691273	−0.345637	−	0.938368i	$-0.612337\pi$
$149$	−1257.27	−0.691273	−0.345637	+	0.938368i	$0.612337\pi$
$150$	380.106	0.206903
$151$	2141.33	1.15403	0.577016	−	0.816733i	$-0.304217\pi$
$151$	2141.33	1.15403	0.577016	+	0.816733i	$0.304217\pi$
$152$	467.745	0.249600
$153$	−360.496	−0.190486
$154$	703.847	0.368296
$155$	1070.63	0.554806
$156$	−982.760	−0.504383
$157$	277.674	0.141152	0.0705759	−	0.997506i	$-0.477516\pi$
$157$	277.674	0.141152	0.0705759	+	0.997506i	$0.477516\pi$
$158$	−1440.17	−0.725148
$159$	2231.03	1.11278
$160$	531.882	0.262806
$161$	−2375.10	−1.16263
$162$	2144.31	1.03995
$163$	1318.04	0.633354	0.316677	−	0.948533i	$-0.397433\pi$
$163$	1318.04	0.633354	0.316677	+	0.948533i	$0.397433\pi$
$164$	−790.859	−0.376559
$165$	−354.869	−0.167434
$166$	−712.463	−0.333120
$167$	513.829	0.238091	0.119046	−	0.992889i	$-0.462016\pi$
$167$	513.829	0.238091	0.119046	+	0.992889i	$0.462016\pi$
$168$	−4313.09	−1.98073
$169$	1677.07	0.763348
$170$	−290.316	−0.130978
$171$	277.979	0.124313
$172$	−936.053	−0.414961
$173$	604.239	0.265546	0.132773	−	0.991146i	$-0.457612\pi$
$173$	604.239	0.265546	0.132773	+	0.991146i	$0.457612\pi$
$174$	3371.82	1.46906
$175$	678.839	0.293231
$176$	422.779	0.181069
$177$	−2056.98	−0.873516
$178$	2356.89	0.992450
$179$	−3174.97	−1.32574	−0.662872	−	0.748733i	$-0.730663\pi$
$179$	−3174.97	−1.32574	−0.662872	+	0.748733i	$0.730663\pi$
$180$	179.014	0.0741272
$181$	161.844	0.0664628	0.0332314	−	0.999448i	$-0.489420\pi$
$181$	161.844	0.0664628	0.0332314	+	0.999448i	$0.489420\pi$
$182$	3982.63	1.62204
$183$	2975.20	1.20182
$184$	−2153.33	−0.862747
$185$	961.652	0.382173
$186$	−3255.62	−1.28340
$187$	271.041	0.105992
$188$	31.4965	0.0122187
$189$	2167.13	0.834052
$190$	223.863	0.0854775
$191$	−1107.93	−0.419721	−0.209860	−	0.977731i	$-0.567301\pi$
$191$	−1107.93	−0.419721	−0.209860	+	0.977731i	$0.567301\pi$
$192$	−3601.26	−1.35364
$193$	361.856	0.134958	0.0674792	−	0.997721i	$-0.478504\pi$
$193$	361.856	0.134958	0.0674792	+	0.997721i	$0.478504\pi$
$194$	1413.48	0.523102
$195$	−2007.98	−0.737408
$196$	−964.945	−0.351656
$197$	1824.88	0.659987	0.329994	−	0.943983i	$-0.392953\pi$
$197$	1824.88	0.659987	0.329994	+	0.943983i	$0.392953\pi$
$198$	379.236	0.136117
$199$	5309.43	1.89133	0.945667	−	0.325136i	$-0.105410\pi$
$199$	5309.43	1.89133	0.945667	+	0.325136i	$0.105410\pi$
$200$	615.454	0.217596
$201$	−5605.62	−1.96711
$202$	3764.24	1.31114
$203$	6021.81	2.08201
$204$	−389.050	−0.133524
$205$	−1615.89	−0.550529
$206$	−1525.75	−0.516039
$207$	−1279.71	−0.429692
$208$	2392.24	0.797462
$209$	−209.000	−0.0691714
$210$	−2064.25	−0.678316
$211$	2164.41	0.706181	0.353090	−	0.935589i	$-0.385131\pi$
$211$	2164.41	0.706181	0.353090	+	0.935589i	$0.385131\pi$
$212$	846.168	0.274128
$213$	−3982.35	−1.28106
$214$	−669.400	−0.213828
$215$	−1912.55	−0.606674
$216$	1964.78	0.618919
$217$	−5814.28	−1.81889
$218$	995.102	0.309160
$219$	−7258.66	−2.23970
$220$	−134.592	−0.0412464
$221$	1533.65	0.466807
$222$	−2924.24	−0.884062
$223$	1445.81	0.434163	0.217082	−	0.976153i	$-0.430346\pi$
$223$	1445.81	0.434163	0.217082	+	0.976153i	$0.430346\pi$
$224$	−2888.50	−0.861589
$225$	365.762	0.108374
$226$	−1828.97	−0.538323
$227$	−2914.91	−0.852289	−0.426145	−	0.904655i	$-0.640128\pi$
$227$	−2914.91	−0.852289	−0.426145	+	0.904655i	$0.640128\pi$
$228$	299.997	0.0871394
$229$	2157.41	0.622559	0.311279	−	0.950318i	$-0.399243\pi$
$229$	2157.41	0.622559	0.311279	+	0.950318i	$0.399243\pi$
$230$	−1030.58	−0.295455
$231$	1927.19	0.548918
$232$	5459.54	1.54498
$233$	−6807.59	−1.91408	−0.957038	−	0.289961i	$-0.906358\pi$
$233$	−6807.59	−1.91408	−0.957038	+	0.289961i	$0.906358\pi$
$234$	2145.86	0.599484
$235$	64.3538	0.0178637
$236$	−780.158	−0.215186
$237$	−3943.30	−1.08078
$238$	1576.62	0.429400
$239$	−100.149	−0.0271050	−0.0135525	−	0.999908i	$-0.504314\pi$
$239$	−100.149	−0.0271050	−0.0135525	+	0.999908i	$0.504314\pi$
$240$	−1239.93	−0.333488
$241$	−4266.66	−1.14041	−0.570206	−	0.821502i	$-0.693137\pi$
$241$	−4266.66	−1.14041	−0.570206	+	0.821502i	$0.693137\pi$
$242$	−285.131	−0.0757392
$243$	3716.42	0.981104
$244$	1128.41	0.296062
$245$	−1971.58	−0.514122
$246$	4913.66	1.27351
$247$	−1182.60	−0.304644
$248$	−5271.38	−1.34973
$249$	−1950.78	−0.496490
$250$	294.557	0.0745176
$251$	−3018.35	−0.759029	−0.379514	−	0.925186i	$-0.623909\pi$
$251$	−3018.35	−0.759029	−0.379514	+	0.925186i	$0.623909\pi$
$252$	−972.172	−0.243020
$253$	962.160	0.239093
$254$	3051.05	0.753700
$255$	−794.909	−0.195212
$256$	−3371.23	−0.823053
$257$	5569.13	1.35172	0.675861	−	0.737029i	$-0.263772\pi$
$257$	5569.13	1.35172	0.675861	+	0.737029i	$0.263772\pi$
$258$	5815.77	1.40339
$259$	−5222.46	−1.25293
$260$	−761.573	−0.181657
$261$	3244.58	0.769481
$262$	4310.51	1.01643
$263$	3029.37	0.710262	0.355131	−	0.934817i	$-0.384436\pi$
$263$	3029.37	0.710262	0.355131	+	0.934817i	$0.384436\pi$
$264$	1747.25	0.407332
$265$	1728.90	0.400775
$266$	−1215.74	−0.280231
$267$	6453.35	1.47917
$268$	−2126.06	−0.484589
$269$	−989.772	−0.224340	−0.112170	−	0.993689i	$-0.535780\pi$
$269$	−989.772	−0.224340	−0.112170	+	0.993689i	$0.535780\pi$
$270$	940.345	0.211954
$271$	−2041.10	−0.457520	−0.228760	−	0.973483i	$-0.573467\pi$
$271$	−2041.10	−0.457520	−0.228760	+	0.973483i	$0.573467\pi$
$272$	947.027	0.211110
$273$	10904.8	2.41753
$274$	−2202.05	−0.485514
$275$	−275.000	−0.0603023
$276$	−1381.08	−0.301200
$277$	−2783.55	−0.603781	−0.301890	−	0.953343i	$-0.597618\pi$
$277$	−2783.55	−0.603781	−0.301890	+	0.953343i	$0.597618\pi$
$278$	4738.34	1.02225
$279$	−3132.76	−0.672235
$280$	−3342.36	−0.713371
$281$	−5169.52	−1.09747	−0.548733	−	0.835998i	$-0.684889\pi$
$281$	−5169.52	−1.09747	−0.548733	+	0.835998i	$0.684889\pi$
$282$	−195.690	−0.0413233
$283$	−5328.78	−1.11930	−0.559652	−	0.828728i	$-0.689065\pi$
$283$	−5328.78	−1.11930	−0.559652	+	0.828728i	$0.689065\pi$
$284$	−1510.40	−0.315583
$285$	612.956	0.127398
$286$	−1613.37	−0.333569
$287$	8775.43	1.80487
$288$	−1556.34	−0.318431
$289$	−4305.87	−0.876423
$290$	2612.94	0.529093
$291$	3870.22	0.779644
$292$	−2753.01	−0.551739
$293$	−2129.21	−0.424538	−0.212269	−	0.977211i	$-0.568085\pi$
$293$	−2129.21	−0.424538	−0.212269	+	0.977211i	$0.568085\pi$
$294$	5995.28	1.18929
$295$	−1594.02	−0.314602
$296$	−4734.82	−0.929750
$297$	−877.913	−0.171521
$298$	2962.70	0.575922
$299$	5444.26	1.05301
$300$	394.733	0.0759664
$301$	10386.5	1.98893
$302$	−5045.94	−0.961460
$303$	10306.8	1.95416
$304$	−730.255	−0.137773
$305$	2305.58	0.432843
$306$	849.491	0.158700
$307$	−1764.59	−0.328048	−0.164024	−	0.986456i	$-0.552447\pi$
$307$	−1764.59	−0.328048	−0.164024	+	0.986456i	$0.552447\pi$
$308$	730.933	0.135223
$309$	−4177.64	−0.769118
$310$	−2522.88	−0.462227
$311$	−939.185	−0.171242	−0.0856210	−	0.996328i	$-0.527287\pi$
$311$	−939.185	−0.171242	−0.0856210	+	0.996328i	$0.527287\pi$
$312$	9886.57	1.79396
$313$	−8217.04	−1.48388	−0.741941	−	0.670466i	$-0.766094\pi$
$313$	−8217.04	−1.48388	−0.741941	+	0.670466i	$0.766094\pi$
$314$	−654.326	−0.117598
$315$	−1986.35	−0.355295
$316$	−1495.59	−0.266245
$317$	−10852.0	−1.92274	−0.961369	−	0.275264i	$-0.911235\pi$
$317$	−10852.0	−1.92274	−0.961369	+	0.275264i	$0.911235\pi$
$318$	−5257.31	−0.927092
$319$	−2439.45	−0.428161
$320$	−2790.73	−0.487521
$321$	−1832.87	−0.318695
$322$	5596.80	0.968627
$323$	−468.161	−0.0806476
$324$	2226.82	0.381828
$325$	−1556.05	−0.265582
$326$	−3105.89	−0.527667
$327$	2724.67	0.460779
$328$	7956.04	1.33933
$329$	−349.487	−0.0585649
$330$	836.232	0.139494
$331$	−3752.38	−0.623109	−0.311555	−	0.950228i	$-0.600850\pi$
$331$	−3752.38	−0.623109	−0.311555	+	0.950228i	$0.600850\pi$
$332$	−739.880	−0.122308
$333$	−2813.89	−0.463063
$334$	−1210.81	−0.198362
$335$	−4343.98	−0.708469
$336$	6733.70	1.09331
$337$	6167.86	0.996987	0.498493	−	0.866893i	$-0.333887\pi$
$337$	6167.86	0.996987	0.498493	+	0.866893i	$0.333887\pi$
$338$	−3951.95	−0.635969
$339$	−5007.87	−0.802330
$340$	−301.488	−0.0480896
$341$	2355.38	0.374050
$342$	−655.044	−0.103569
$343$	1393.43	0.219352
$344$	9416.69	1.47591
$345$	−2821.83	−0.440354
$346$	−1423.86	−0.221235
$347$	3774.19	0.583888	0.291944	−	0.956435i	$-0.405698\pi$
$347$	3774.19	0.583888	0.291944	+	0.956435i	$0.405698\pi$
$348$	3501.57	0.539380
$349$	−10385.6	−1.59292	−0.796462	−	0.604689i	$-0.793297\pi$
$349$	−10385.6	−1.59292	−0.796462	+	0.604689i	$0.793297\pi$
$350$	−1599.65	−0.244300
$351$	−4967.55	−0.755409
$352$	1170.14	0.177184
$353$	12255.6	1.84788	0.923940	−	0.382536i	$-0.124949\pi$
$353$	12255.6	1.84788	0.923940	+	0.382536i	$0.124949\pi$
$354$	4847.18	0.727754
$355$	−3086.06	−0.461383
$356$	2447.58	0.364387
$357$	4316.92	0.639988
$358$	7481.66	1.10452
$359$	−11894.1	−1.74859	−0.874295	−	0.485394i	$-0.838676\pi$
$359$	−11894.1	−1.74859	−0.874295	+	0.485394i	$0.838676\pi$
$360$	−1800.88	−0.263652
$361$	361.000	0.0526316
$362$	−381.378	−0.0553723
$363$	−780.712	−0.112884
$364$	4135.89	0.595548
$365$	−5624.97	−0.806643
$366$	−7010.91	−1.00127
$367$	13143.1	1.86938	0.934689	−	0.355466i	$-0.115678\pi$
$367$	13143.1	1.86938	0.934689	+	0.355466i	$0.115678\pi$
$368$	3361.83	0.476216
$369$	4728.24	0.667053
$370$	−2266.09	−0.318401
$371$	−9389.14	−1.31391
$372$	−3380.90	−0.471213
$373$	11525.4	1.59989	0.799947	−	0.600071i	$-0.204861\pi$
$373$	11525.4	1.59989	0.799947	+	0.600071i	$0.204861\pi$
$374$	−638.694	−0.0883051
$375$	806.521	0.111063
$376$	−316.855	−0.0434589
$377$	−13803.3	−1.88570
$378$	−5106.75	−0.694875
$379$	1313.56	0.178029	0.0890145	−	0.996030i	$-0.471628\pi$
$379$	1313.56	0.178029	0.0890145	+	0.996030i	$0.471628\pi$
$380$	232.478	0.0313838
$381$	8354.04	1.12333
$382$	2610.77	0.349683
$383$	−3728.61	−0.497450	−0.248725	−	0.968574i	$-0.580011\pi$
$383$	−3728.61	−0.497450	−0.248725	+	0.968574i	$0.580011\pi$
$384$	2995.32	0.398058
$385$	1493.45	0.197696
$386$	−852.696	−0.112438
$387$	5596.30	0.735080
$388$	1467.87	0.192062
$389$	2863.51	0.373229	0.186614	−	0.982433i	$-0.440249\pi$
$389$	2863.51	0.373229	0.186614	+	0.982433i	$0.440249\pi$
$390$	4731.71	0.614358
$391$	2155.24	0.278760
$392$	9707.35	1.25075
$393$	11802.5	1.51491
$394$	−4300.25	−0.549856
$395$	−3055.79	−0.389250
$396$	393.830	0.0499765
$397$	−3984.06	−0.503663	−0.251831	−	0.967771i	$-0.581033\pi$
$397$	−3984.06	−0.503663	−0.251831	+	0.967771i	$0.581033\pi$
$398$	−12511.4	−1.57573
$399$	−3328.79	−0.417664
$400$	−960.862	−0.120108
$401$	−3630.88	−0.452164	−0.226082	−	0.974108i	$-0.572592\pi$
$401$	−3630.88	−0.452164	−0.226082	+	0.974108i	$0.572592\pi$
$402$	13209.4	1.63887
$403$	13327.6	1.64739
$404$	3909.09	0.481398
$405$	4549.86	0.558233
$406$	−14190.1	−1.73459
$407$	2115.63	0.257661
$408$	3913.84	0.474912
$409$	1002.80	0.121236	0.0606179	−	0.998161i	$-0.480693\pi$
$409$	1002.80	0.121236	0.0606179	+	0.998161i	$0.480693\pi$
$410$	3807.76	0.458663
$411$	−6029.41	−0.723623
$412$	−1584.46	−0.189468
$413$	8656.69	1.03140
$414$	3015.59	0.357990
$415$	−1511.73	−0.178814
$416$	6621.08	0.780349
$417$	12974.0	1.52359
$418$	492.498	0.0576289
$419$	−9732.69	−1.13478	−0.567390	−	0.823449i	$-0.692047\pi$
$419$	−9732.69	−1.13478	−0.567390	+	0.823449i	$0.692047\pi$
$420$	−2143.68	−0.249050
$421$	8011.74	0.927478	0.463739	−	0.885972i	$-0.346508\pi$
$421$	8011.74	0.927478	0.463739	+	0.885972i	$0.346508\pi$
$422$	−5100.33	−0.588342
$423$	−188.305	−0.0216447
$424$	−8512.45	−0.975003
$425$	−616.001	−0.0703070
$426$	9384.22	1.06729
$427$	−12520.9	−1.41904
$428$	−695.160	−0.0785089
$429$	−4417.56	−0.497160
$430$	4506.83	0.505439
$431$	1477.76	0.165154	0.0825770	−	0.996585i	$-0.473685\pi$
$431$	1477.76	0.165154	0.0825770	+	0.996585i	$0.473685\pi$
$432$	−3067.46	−0.341628
$433$	3804.42	0.422237	0.211118	−	0.977460i	$-0.432289\pi$
$433$	3804.42	0.422237	0.211118	+	0.977460i	$0.432289\pi$
$434$	13701.1	1.51537
$435$	7154.44	0.788573
$436$	1033.40	0.113511
$437$	−1661.91	−0.181922
$438$	17104.7	1.86597
$439$	4583.66	0.498328	0.249164	−	0.968461i	$-0.419844\pi$
$439$	4583.66	0.498328	0.249164	+	0.968461i	$0.419844\pi$
$440$	1354.00	0.146703
$441$	5769.04	0.622939
$442$	−3613.97	−0.388912
$443$	−1021.15	−0.109517	−0.0547587	−	0.998500i	$-0.517439\pi$
$443$	−1021.15	−0.109517	−0.0547587	+	0.998500i	$0.517439\pi$
$444$	−3036.77	−0.324591
$445$	5000.92	0.532733
$446$	−3406.97	−0.361715
$447$	8112.13	0.858368
$448$	15155.7	1.59830
$449$	8234.97	0.865551	0.432775	−	0.901502i	$-0.357534\pi$
$449$	8234.97	0.865551	0.432775	+	0.901502i	$0.357534\pi$
$450$	−861.900	−0.0902897
$451$	−3554.95	−0.371167
$452$	−1899.35	−0.197650
$453$	−13816.2	−1.43298
$454$	6868.85	0.710069
$455$	8450.47	0.870691
$456$	−3017.97	−0.309933
$457$	−11164.0	−1.14274	−0.571368	−	0.820694i	$-0.693587\pi$
$457$	−11164.0	−1.14274	−0.571368	+	0.820694i	$0.693587\pi$
$458$	−5083.84	−0.518673
$459$	−1966.53	−0.199978
$460$	−1070.24	−0.108479
$461$	−5003.36	−0.505487	−0.252744	−	0.967533i	$-0.581333\pi$
$461$	−5003.36	−0.505487	−0.252744	+	0.967533i	$0.581333\pi$
$462$	−4541.34	−0.457321
$463$	−3650.73	−0.366444	−0.183222	−	0.983072i	$-0.558653\pi$
$463$	−3650.73	−0.366444	−0.183222	+	0.983072i	$0.558653\pi$
$464$	−8523.56	−0.852794
$465$	−6907.87	−0.688914
$466$	16041.8	1.59468
$467$	−17857.9	−1.76951	−0.884757	−	0.466052i	$-0.845676\pi$
$467$	−17857.9	−1.76951	−0.884757	+	0.466052i	$0.845676\pi$
$468$	2228.44	0.220106
$469$	23590.9	2.32266
$470$	−151.647	−0.0148828
$471$	−1791.60	−0.175271
$472$	7848.39	0.765364
$473$	−4207.61	−0.409019
$474$	9292.19	0.900432
$475$	475.000	0.0458831
$476$	1637.29	0.157658
$477$	−5058.92	−0.485601
$478$	235.996	0.0225820
$479$	−9206.91	−0.878235	−0.439117	−	0.898430i	$-0.644709\pi$
$479$	−9206.91	−0.878235	−0.439117	+	0.898430i	$0.644709\pi$
$480$	−3431.79	−0.326331
$481$	11971.0	1.13479
$482$	10054.2	0.950114
$483$	15324.5	1.44367
$484$	−296.103	−0.0278083
$485$	2999.17	0.280794
$486$	−8757.56	−0.817389
$487$	−11740.6	−1.09244	−0.546219	−	0.837643i	$-0.683933\pi$
$487$	−11740.6	−1.09244	−0.546219	+	0.837643i	$0.683933\pi$
$488$	−11351.8	−1.05302
$489$	−8504.20	−0.786449
$490$	4645.94	0.428331
$491$	−14567.3	−1.33893	−0.669463	−	0.742845i	$-0.733476\pi$
$491$	−14567.3	−1.33893	−0.669463	+	0.742845i	$0.733476\pi$
$492$	5102.75	0.467581
$493$	−5464.39	−0.499196
$494$	2786.74	0.253808
$495$	804.676	0.0730657
$496$	8229.81	0.745019
$497$	16759.5	1.51261
$498$	4596.93	0.413641
$499$	−5091.36	−0.456754	−0.228377	−	0.973573i	$-0.573342\pi$
$499$	−5091.36	−0.456754	−0.228377	+	0.973573i	$0.573342\pi$
$500$	305.892	0.0273598
$501$	−3315.31	−0.295643
$502$	7112.59	0.632371
$503$	−13576.4	−1.20346	−0.601729	−	0.798700i	$-0.705521\pi$
$503$	−13576.4	−1.20346	−0.601729	+	0.798700i	$0.705521\pi$
$504$	9780.05	0.864361
$505$	7987.09	0.703804
$506$	−2267.28	−0.199196
$507$	−10820.8	−0.947864
$508$	3168.46	0.276728
$509$	−2925.63	−0.254766	−0.127383	−	0.991854i	$-0.540658\pi$
$509$	−2925.63	−0.254766	−0.127383	+	0.991854i	$0.540658\pi$
$510$	1873.16	0.162638
$511$	30547.6	2.64452
$512$	11658.0	1.00628
$513$	1516.39	0.130508
$514$	−13123.4	−1.12616
$515$	−3237.39	−0.277003
$516$	6039.57	0.515266
$517$	141.578	0.0120437
$518$	12306.5	1.04385
$519$	−3898.65	−0.329734
$520$	7661.43	0.646107
$521$	−11989.9	−1.00823	−0.504113	−	0.863638i	$-0.668181\pi$
$521$	−11989.9	−1.00823	−0.504113	+	0.863638i	$0.668181\pi$
$522$	−7645.70	−0.641079
$523$	−12968.7	−1.08429	−0.542143	−	0.840286i	$-0.682387\pi$
$523$	−12968.7	−1.08429	−0.542143	+	0.840286i	$0.682387\pi$
$524$	4476.39	0.373191
$525$	−4379.99	−0.364111
$526$	−7138.56	−0.591742
$527$	5276.07	0.436108
$528$	−2727.84	−0.224837
$529$	−4516.16	−0.371181
$530$	−4074.06	−0.333898
$531$	4664.27	0.381190
$532$	−1262.52	−0.102889
$533$	−20115.2	−1.63469
$534$	−15207.0	−1.23234
$535$	−1420.36	−0.114780
$536$	21388.2	1.72356
$537$	20485.4	1.64620
$538$	2332.35	0.186905
$539$	−4337.48	−0.346621
$540$	976.532	0.0778208
$541$	5975.37	0.474863	0.237432	−	0.971404i	$-0.423694\pi$
$541$	5975.37	0.474863	0.237432	+	0.971404i	$0.423694\pi$
$542$	4809.75	0.381174
$543$	−1044.24	−0.0825282
$544$	2621.12	0.206580
$545$	2111.44	0.165953
$546$	−25696.6	−2.01412
$547$	−9267.52	−0.724407	−0.362204	−	0.932099i	$-0.617975\pi$
$547$	−9267.52	−0.724407	−0.362204	+	0.932099i	$0.617975\pi$
$548$	−2286.79	−0.178261
$549$	−6746.35	−0.524457
$550$	648.024	0.0502397
$551$	4213.60	0.325781
$552$	13893.6	1.07129
$553$	16595.1	1.27612
$554$	6559.30	0.503029
$555$	−6204.74	−0.474552
$556$	4920.68	0.375330
$557$	7898.82	0.600868	0.300434	−	0.953803i	$-0.402868\pi$
$557$	7898.82	0.600868	0.300434	+	0.953803i	$0.402868\pi$
$558$	7382.20	0.560060
$559$	−23808.2	−1.80140
$560$	5218.17	0.393764
$561$	−1748.80	−0.131612
$562$	12181.7	0.914333
$563$	18906.6	1.41530	0.707652	−	0.706561i	$-0.249754\pi$
$563$	18906.6	1.41530	0.707652	+	0.706561i	$0.249754\pi$
$564$	−203.221	−0.0151722
$565$	−3880.76	−0.288964
$566$	12557.0	0.932527
$567$	−24709.0	−1.83012
$568$	15194.6	1.12245
$569$	−714.759	−0.0526612	−0.0263306	−	0.999653i	$-0.508382\pi$
$569$	−714.759	−0.0526612	−0.0263306	+	0.999653i	$0.508382\pi$
$570$	−1444.40	−0.106139
$571$	−5519.03	−0.404491	−0.202246	−	0.979335i	$-0.564824\pi$
$571$	−5519.03	−0.404491	−0.202246	+	0.979335i	$0.564824\pi$
$572$	−1675.46	−0.122473
$573$	7148.52	0.521176
$574$	−20678.9	−1.50369
$575$	−2186.73	−0.158596
$576$	8165.95	0.590708
$577$	9321.44	0.672542	0.336271	−	0.941765i	$-0.390834\pi$
$577$	9321.44	0.672542	0.336271	+	0.941765i	$0.390834\pi$
$578$	10146.6	0.730176
$579$	−2334.75	−0.167580
$580$	2713.49	0.194261
$581$	8209.76	0.586228
$582$	−9120.00	−0.649546
$583$	3803.57	0.270202
$584$	27695.3	1.96240
$585$	4553.16	0.321795
$586$	5017.37	0.353696
$587$	−10907.6	−0.766959	−0.383480	−	0.923549i	$-0.625274\pi$
$587$	−10907.6	−0.766959	−0.383480	+	0.923549i	$0.625274\pi$
$588$	6225.99	0.436659
$589$	−4068.39	−0.284610
$590$	3756.24	0.262105
$591$	−11774.5	−0.819520
$592$	7392.12	0.513200
$593$	−23021.6	−1.59424	−0.797118	−	0.603823i	$-0.793643\pi$
$593$	−23021.6	−1.59424	−0.797118	+	0.603823i	$0.793643\pi$
$594$	2068.76	0.142899
$595$	3345.33	0.230496
$596$	3076.71	0.211455
$597$	−34257.3	−2.34851
$598$	−12829.1	−0.877295
$599$	282.997	0.0193038	0.00965188	−	0.999953i	$-0.496928\pi$
$599$	282.997	0.0193038	0.00965188	+	0.999953i	$0.496928\pi$
$600$	−3971.01	−0.270193
$601$	13602.3	0.923209	0.461604	−	0.887086i	$-0.347274\pi$
$601$	13602.3	0.923209	0.461604	+	0.887086i	$0.347274\pi$
$602$	−24475.3	−1.65704
$603$	12710.9	0.858421
$604$	−5240.11	−0.353009
$605$	−605.000	−0.0406558
$606$	−24287.5	−1.62807
$607$	837.322	0.0559898	0.0279949	−	0.999608i	$-0.491088\pi$
$607$	837.322	0.0559898	0.0279949	+	0.999608i	$0.491088\pi$
$608$	−2021.15	−0.134817
$609$	−38853.7	−2.58527
$610$	−5432.99	−0.360615
$611$	801.103	0.0530428
$612$	882.181	0.0582681
$613$	13400.7	0.882951	0.441476	−	0.897273i	$-0.354455\pi$
$613$	13400.7	0.882951	0.441476	+	0.897273i	$0.354455\pi$
$614$	4158.18	0.273307
$615$	10426.0	0.683603
$616$	−7353.18	−0.480955
$617$	−13724.1	−0.895478	−0.447739	−	0.894164i	$-0.647771\pi$
$617$	−13724.1	−0.895478	−0.447739	+	0.894164i	$0.647771\pi$
$618$	9844.40	0.640776
$619$	−28590.7	−1.85647	−0.928235	−	0.371993i	$-0.878674\pi$
$619$	−28590.7	−1.85647	−0.928235	+	0.371993i	$0.878674\pi$
$620$	−2619.97	−0.169711
$621$	−6980.93	−0.451103
$622$	2213.14	0.142667
$623$	−27158.6	−1.74652
$624$	−15435.1	−0.990225
$625$	625.000	0.0400000
$626$	19363.1	1.23627
$627$	1348.50	0.0858916
$628$	−679.506	−0.0431771
$629$	4739.03	0.300410
$630$	4680.74	0.296008
$631$	−14846.0	−0.936625	−0.468312	−	0.883563i	$-0.655138\pi$
$631$	−14846.0	−0.936625	−0.468312	+	0.883563i	$0.655138\pi$
$632$	15045.6	0.946965
$633$	−13965.1	−0.876879
$634$	25572.2	1.60189
$635$	6473.82	0.404576
$636$	−5459.62	−0.340390
$637$	−24543.1	−1.52658
$638$	5748.46	0.356714
$639$	9030.10	0.559038
$640$	2321.17	0.143363
$641$	−24225.6	−1.49275	−0.746376	−	0.665525i	$-0.768208\pi$
$641$	−24225.6	−1.49275	−0.746376	+	0.665525i	$0.768208\pi$
$642$	4319.08	0.265515
$643$	−11195.5	−0.686637	−0.343318	−	0.939219i	$-0.611551\pi$
$643$	−11195.5	−0.686637	−0.343318	+	0.939219i	$0.611551\pi$
$644$	5812.18	0.355640
$645$	12340.1	0.753319
$646$	1103.20	0.0671901
$647$	28046.4	1.70420	0.852101	−	0.523378i	$-0.175328\pi$
$647$	28046.4	1.70420	0.852101	+	0.523378i	$0.175328\pi$
$648$	−22401.8	−1.35807
$649$	−3506.85	−0.212105
$650$	3666.76	0.221265
$651$	37514.7	2.25855
$652$	−3225.41	−0.193738
$653$	22676.8	1.35898	0.679489	−	0.733686i	$-0.262202\pi$
$653$	22676.8	1.35898	0.679489	+	0.733686i	$0.262202\pi$
$654$	−6420.56	−0.383890
$655$	9146.19	0.545605
$656$	−12421.2	−0.739276
$657$	16459.2	0.977374
$658$	823.550	0.0487923
$659$	12932.9	0.764481	0.382240	−	0.924063i	$-0.375153\pi$
$659$	12932.9	0.764481	0.382240	+	0.924063i	$0.375153\pi$
$660$	868.412	0.0512165
$661$	12825.5	0.754698	0.377349	−	0.926071i	$-0.376836\pi$
$661$	12825.5	0.754698	0.377349	+	0.926071i	$0.376836\pi$
$662$	8842.29	0.519132
$663$	−9895.35	−0.579644
$664$	7443.20	0.435018
$665$	−2579.59	−0.150424
$666$	6630.79	0.385792
$667$	−19397.9	−1.12607
$668$	−1257.41	−0.0728302
$669$	−9328.59	−0.539109
$670$	10236.4	0.590248
$671$	5072.28	0.291823
$672$	18637.1	1.06985
$673$	3363.23	0.192634	0.0963171	−	0.995351i	$-0.469294\pi$
$673$	3363.23	0.192634	0.0963171	+	0.995351i	$0.469294\pi$
$674$	−14534.3	−0.830621
$675$	1995.26	0.113774
$676$	−4104.03	−0.233502
$677$	−22540.1	−1.27960	−0.639798	−	0.768543i	$-0.720982\pi$
$677$	−22540.1	−1.27960	−0.639798	+	0.768543i	$0.720982\pi$
$678$	11800.8	0.668447
$679$	−16287.6	−0.920561
$680$	3032.97	0.171043
$681$	18807.5	1.05830
$682$	−5550.35	−0.311633
$683$	6738.98	0.377540	0.188770	−	0.982021i	$-0.439550\pi$
$683$	6738.98	0.377540	0.188770	+	0.982021i	$0.439550\pi$
$684$	−680.252	−0.0380264
$685$	−4672.39	−0.260617
$686$	−3283.54	−0.182749
$687$	−13920.0	−0.773044
$688$	−14701.6	−0.814669
$689$	21522.0	1.19002
$690$	6649.50	0.366873
$691$	35040.3	1.92908	0.964540	−	0.263937i	$-0.0850211\pi$
$691$	35040.3	1.92908	0.964540	+	0.263937i	$0.0850211\pi$
$692$	−1478.65	−0.0812283
$693$	−4369.97	−0.239540
$694$	−8893.70	−0.486456
$695$	10054.0	0.548732
$696$	−35225.8	−1.91844
$697$	−7963.11	−0.432747
$698$	24473.3	1.32711
$699$	43923.7	2.37675
$700$	−1661.21	−0.0896969
$701$	−4384.32	−0.236225	−0.118112	−	0.993000i	$-0.537684\pi$
$701$	−4384.32	−0.236225	−0.118112	+	0.993000i	$0.537684\pi$
$702$	11705.8	0.629355
$703$	−3654.28	−0.196051
$704$	−6139.61	−0.328687
$705$	−415.222	−0.0221818
$706$	−28879.8	−1.53953
$707$	−43375.6	−2.30737
$708$	5033.71	0.267201
$709$	22981.5	1.21733	0.608667	−	0.793426i	$-0.291705\pi$
$709$	22981.5	1.21733	0.608667	+	0.793426i	$0.291705\pi$
$710$	7272.15	0.384393
$711$	8941.54	0.471637
$712$	−24622.7	−1.29603
$713$	18729.4	0.983760
$714$	−10172.6	−0.533194
$715$	−3423.31	−0.179055
$716$	7769.57	0.405534
$717$	646.178	0.0336568
$718$	28027.8	1.45681
$719$	−11512.5	−0.597140	−0.298570	−	0.954388i	$-0.596510\pi$
$719$	−11512.5	−0.597140	−0.298570	+	0.954388i	$0.596510\pi$
$720$	2811.57	0.145529
$721$	17581.3	0.908132
$722$	−850.679	−0.0438490
$723$	27529.2	1.41607
$724$	−396.054	−0.0203304
$725$	5544.21	0.284010
$726$	1839.71	0.0940470
$727$	−9327.49	−0.475842	−0.237921	−	0.971284i	$-0.576466\pi$
$727$	−9327.49	−0.475842	−0.237921	+	0.971284i	$0.576466\pi$
$728$	−41607.0	−2.11821
$729$	590.303	0.0299905
$730$	13255.0	0.672040
$731$	−9425.07	−0.476879
$732$	−7280.71	−0.367627
$733$	38033.1	1.91649	0.958243	−	0.285954i	$-0.0923105\pi$
$733$	38033.1	1.91649	0.958243	+	0.285954i	$0.0923105\pi$
$734$	−30971.0	−1.55744
$735$	12721.0	0.638395
$736$	9304.64	0.465997
$737$	−9556.76	−0.477650
$738$	−11141.9	−0.555743
$739$	−20563.0	−1.02357	−0.511787	−	0.859113i	$-0.671016\pi$
$739$	−20563.0	−1.02357	−0.511787	+	0.859113i	$0.671016\pi$
$740$	−2353.29	−0.116904
$741$	7630.33	0.378282
$742$	22125.1	1.09466
$743$	16461.0	0.812781	0.406390	−	0.913700i	$-0.366787\pi$
$743$	16461.0	0.812781	0.406390	+	0.913700i	$0.366787\pi$
$744$	34011.8	1.67599
$745$	6286.36	0.309147
$746$	−27158.9	−1.33292
$747$	4423.46	0.216661
$748$	−663.273	−0.0324220
$749$	7713.55	0.376297
$750$	−1900.53	−0.0925300
$751$	−20471.2	−0.994681	−0.497341	−	0.867555i	$-0.665690\pi$
$751$	−20471.2	−0.994681	−0.497341	+	0.867555i	$0.665690\pi$
$752$	494.681	0.0239882
$753$	19474.9	0.942502
$754$	32526.9	1.57103
$755$	−10706.6	−0.516099
$756$	−5303.26	−0.255129
$757$	−25570.9	−1.22773	−0.613863	−	0.789412i	$-0.710385\pi$
$757$	−25570.9	−1.22773	−0.613863	+	0.789412i	$0.710385\pi$
$758$	−3095.34	−0.148322
$759$	−6208.02	−0.296886
$760$	−2338.73	−0.111624
$761$	−29336.6	−1.39744	−0.698718	−	0.715397i	$-0.746246\pi$
$761$	−29336.6	−1.39744	−0.698718	+	0.715397i	$0.746246\pi$
$762$	−19685.9	−0.935885
$763$	−11466.6	−0.544063
$764$	2711.24	0.128389
$765$	1802.48	0.0851879
$766$	8786.30	0.414441
$767$	−19843.1	−0.934149
$768$	21751.7	1.02200
$769$	14454.2	0.677804	0.338902	−	0.940822i	$-0.389944\pi$
$769$	14454.2	0.677804	0.338902	+	0.940822i	$0.389944\pi$
$770$	−3519.24	−0.164707
$771$	−35932.9	−1.67846
$772$	−885.510	−0.0412826
$773$	−39749.0	−1.84951	−0.924754	−	0.380564i	$-0.875730\pi$
$773$	−39749.0	−1.84951	−0.924754	+	0.380564i	$0.875730\pi$
$774$	−13187.4	−0.612419
$775$	−5353.14	−0.248117
$776$	−14766.8	−0.683115
$777$	33696.2	1.55578
$778$	−6747.73	−0.310949
$779$	6140.37	0.282416
$780$	4913.80	0.225567
$781$	−6789.33	−0.311064
$782$	−5078.73	−0.232244
$783$	17699.4	0.807823
$784$	−15155.3	−0.690386
$785$	−1388.37	−0.0631250
$786$	−27812.1	−1.26212
$787$	−27724.5	−1.25574	−0.627872	−	0.778317i	$-0.716074\pi$
$787$	−27724.5	−1.25574	−0.627872	+	0.778317i	$0.716074\pi$
$788$	−4465.73	−0.201885
$789$	−19546.0	−0.881947
$790$	7200.83	0.324296
$791$	21075.3	0.947347
$792$	−3961.93	−0.177754
$793$	28700.8	1.28524
$794$	9388.24	0.419618
$795$	−11155.1	−0.497650
$796$	−12992.9	−0.578544
$797$	−5205.55	−0.231355	−0.115678	−	0.993287i	$-0.536904\pi$
$797$	−5205.55	−0.231355	−0.115678	+	0.993287i	$0.536904\pi$
$798$	7844.13	0.347969
$799$	317.136	0.0140419
$800$	−2659.41	−0.117530
$801$	−14633.2	−0.645490
$802$	8556.00	0.376712
$803$	−12374.9	−0.543838
$804$	13717.7	0.601724
$805$	11875.5	0.519945
$806$	−31405.9	−1.37249
$807$	6386.17	0.278567
$808$	−39325.5	−1.71221
$809$	37668.8	1.63704	0.818520	−	0.574478i	$-0.194795\pi$
$809$	37668.8	1.63704	0.818520	+	0.574478i	$0.194795\pi$
$810$	−10721.5	−0.465082
$811$	−2340.35	−0.101333	−0.0506664	−	0.998716i	$-0.516135\pi$
$811$	−2340.35	−0.101333	−0.0506664	+	0.998716i	$0.516135\pi$
$812$	−14736.2	−0.636870
$813$	13169.5	0.568112
$814$	−4985.39	−0.214666
$815$	−6590.19	−0.283245
$816$	−6110.38	−0.262140
$817$	7267.69	0.311217
$818$	−2363.06	−0.101005
$819$	−24726.9	−1.05498
$820$	3954.29	0.168402
$821$	−40055.3	−1.70273	−0.851364	−	0.524575i	$-0.824224\pi$
$821$	−40055.3	−1.70273	−0.851364	+	0.524575i	$0.824224\pi$
$822$	14208.0	0.602873
$823$	33627.2	1.42427	0.712133	−	0.702044i	$-0.247729\pi$
$823$	33627.2	1.42427	0.712133	+	0.702044i	$0.247729\pi$
$824$	15939.7	0.673891
$825$	1774.35	0.0748786
$826$	−20399.1	−0.859292
$827$	−3048.91	−0.128200	−0.0640998	−	0.997943i	$-0.520418\pi$
$827$	−3048.91	−0.128200	−0.0640998	+	0.997943i	$0.520418\pi$
$828$	3131.63	0.131439
$829$	43131.8	1.80703	0.903516	−	0.428554i	$-0.140977\pi$
$829$	43131.8	1.80703	0.903516	+	0.428554i	$0.140977\pi$
$830$	3562.31	0.148976
$831$	17959.9	0.749727
$832$	−34740.2	−1.44760
$833$	−9715.98	−0.404128
$834$	−30572.6	−1.26935
$835$	−2569.14	−0.106478
$836$	511.451	0.0211590
$837$	−17089.4	−0.705731
$838$	22934.6	0.945422
$839$	23686.8	0.974684	0.487342	−	0.873211i	$-0.337967\pi$
$839$	23686.8	0.974684	0.487342	+	0.873211i	$0.337967\pi$
$840$	21565.4	0.885808
$841$	24792.3	1.01654
$842$	−18879.3	−0.772711
$843$	33354.6	1.36274
$844$	−5296.60	−0.216015
$845$	−8385.37	−0.341379
$846$	443.733	0.0180329
$847$	3285.58	0.133287
$848$	13289.8	0.538178
$849$	34382.2	1.38986
$850$	1451.58	0.0585749
$851$	16823.0	0.677654
$852$	9745.35	0.391866
$853$	29848.4	1.19811	0.599057	−	0.800707i	$-0.295542\pi$
$853$	29848.4	1.19811	0.599057	+	0.800707i	$0.295542\pi$
$854$	29505.0	1.18225
$855$	−1389.90	−0.0555946
$856$	6993.31	0.279237
$857$	−43619.9	−1.73866	−0.869329	−	0.494235i	$-0.835448\pi$
$857$	−43619.9	−1.73866	−0.869329	+	0.494235i	$0.835448\pi$
$858$	10409.8	0.414200
$859$	−25254.6	−1.00312	−0.501558	−	0.865124i	$-0.667240\pi$
$859$	−25254.6	−1.00312	−0.501558	+	0.865124i	$0.667240\pi$
$860$	4680.26	0.185576
$861$	−56620.5	−2.24114
$862$	−3482.28	−0.137595
$863$	21076.8	0.831358	0.415679	−	0.909511i	$-0.363544\pi$
$863$	21076.8	0.831358	0.415679	+	0.909511i	$0.363544\pi$
$864$	−8489.92	−0.334298
$865$	−3021.20	−0.118756
$866$	−8964.93	−0.351779
$867$	27782.2	1.08827
$868$	14228.3	0.556383
$869$	−6722.74	−0.262432
$870$	−16859.1	−0.656985
$871$	−54075.7	−2.10366
$872$	−10396.0	−0.403729
$873$	−8775.84	−0.340226
$874$	3916.22	0.151565
$875$	−3394.20	−0.131137
$876$	17762.9	0.685106
$877$	10060.5	0.387366	0.193683	−	0.981064i	$-0.437957\pi$
$877$	10060.5	0.387366	0.193683	+	0.981064i	$0.437957\pi$
$878$	−10801.2	−0.415173
$879$	13738.0	0.527157
$880$	−2113.90	−0.0809766
$881$	−35857.7	−1.37126	−0.685628	−	0.727952i	$-0.740472\pi$
$881$	−35857.7	−1.37126	−0.685628	+	0.727952i	$0.740472\pi$
$882$	−13594.5	−0.518990
$883$	22664.3	0.863775	0.431888	−	0.901927i	$-0.357848\pi$
$883$	22664.3	0.863775	0.431888	+	0.901927i	$0.357848\pi$
$884$	−3753.04	−0.142792
$885$	10284.9	0.390648
$886$	2406.29	0.0912424
$887$	−4504.72	−0.170523	−0.0852615	−	0.996359i	$-0.527173\pi$
$887$	−4504.72	−0.170523	−0.0852615	+	0.996359i	$0.527173\pi$
$888$	30549.9	1.15449
$889$	−35157.5	−1.32637
$890$	−11784.4	−0.443837
$891$	10009.7	0.376361
$892$	−3538.08	−0.132807
$893$	−244.544	−0.00916391
$894$	−19115.8	−0.715134
$895$	15874.8	0.592891
$896$	−12605.6	−0.470005
$897$	−35127.3	−1.30754
$898$	−19405.3	−0.721118
$899$	−47486.3	−1.76169
$900$	−895.068	−0.0331507
$901$	8520.02	0.315031
$902$	8377.08	0.309231
$903$	−67015.5	−2.46970
$904$	19107.5	0.702992
$905$	−809.220	−0.0297231
$906$	32557.2	1.19386
$907$	−986.035	−0.0360979	−0.0180489	−	0.999837i	$-0.505745\pi$
$907$	−986.035	−0.0360979	−0.0180489	+	0.999837i	$0.505745\pi$
$908$	7133.18	0.260708
$909$	−23371.0	−0.852769
$910$	−19913.1	−0.725400
$911$	−10722.4	−0.389954	−0.194977	−	0.980808i	$-0.562463\pi$
$911$	−10722.4	−0.389954	−0.194977	+	0.980808i	$0.562463\pi$
$912$	4711.73	0.171076
$913$	−3325.80	−0.120556
$914$	26307.5	0.952050
$915$	−14876.0	−0.537470
$916$	−5279.48	−0.190436
$917$	−49670.3	−1.78872
$918$	4634.03	0.166608
$919$	−32685.8	−1.17324	−0.586618	−	0.809864i	$-0.699541\pi$
$919$	−32685.8	−1.17324	−0.586618	+	0.809864i	$0.699541\pi$
$920$	10766.6	0.385832
$921$	11385.5	0.407344
$922$	11790.2	0.421137
$923$	−38416.5	−1.36998
$924$	−4716.10	−0.167909
$925$	−4808.26	−0.170913
$926$	8602.76	0.305296
$927$	9472.91	0.335632
$928$	−23590.9	−0.834494
$929$	9821.75	0.346869	0.173434	−	0.984845i	$-0.444514\pi$
$929$	9821.75	0.346869	0.173434	+	0.984845i	$0.444514\pi$
$930$	16278.1	0.573956
$931$	7492.01	0.263739
$932$	16659.1	0.585500
$933$	6059.78	0.212635
$934$	42081.2	1.47424
$935$	−1355.20	−0.0474009
$936$	−22418.1	−0.782861
$937$	−19102.5	−0.666010	−0.333005	−	0.942925i	$-0.608063\pi$
$937$	−19102.5	−0.666010	−0.333005	+	0.942925i	$0.608063\pi$
$938$	−55590.9	−1.93508
$939$	53017.8	1.84257
$940$	−157.482	−0.00546437
$941$	33942.0	1.17585	0.587926	−	0.808915i	$-0.299945\pi$
$941$	33942.0	1.17585	0.587926	+	0.808915i	$0.299945\pi$
$942$	4221.82	0.146024
$943$	−28268.0	−0.976176
$944$	−12253.1	−0.422462
$945$	−10835.7	−0.372999
$946$	9915.03	0.340767
$947$	12962.3	0.444792	0.222396	−	0.974956i	$-0.428612\pi$
$947$	12962.3	0.444792	0.222396	+	0.974956i	$0.428612\pi$
$948$	9649.78	0.330601
$949$	−70022.0	−2.39516
$950$	−1119.31	−0.0382267
$951$	70018.8	2.38750
$952$	−16471.2	−0.560750
$953$	33850.0	1.15059	0.575293	−	0.817948i	$-0.304888\pi$
$953$	33850.0	1.15059	0.575293	+	0.817948i	$0.304888\pi$
$954$	11921.1	0.404570
$955$	5539.63	0.187705
$956$	245.078	0.00829120
$957$	15739.8	0.531656
$958$	21695.6	0.731685
$959$	25374.4	0.854414
$960$	18006.3	0.605365
$961$	16058.8	0.539049
$962$	−28209.2	−0.945427
$963$	4156.10	0.139074
$964$	10441.1	0.348843
$965$	−1809.28	−0.0603552
$966$	−36111.5	−1.20276
$967$	12019.4	0.399708	0.199854	−	0.979826i	$-0.435953\pi$
$967$	12019.4	0.399708	0.199854	+	0.979826i	$0.435953\pi$
$968$	2978.80	0.0989072
$969$	3020.65	0.100142
$970$	−7067.39	−0.233938
$971$	−22215.6	−0.734226	−0.367113	−	0.930176i	$-0.619654\pi$
$971$	−22215.6	−0.734226	−0.367113	+	0.930176i	$0.619654\pi$
$972$	−9094.57	−0.300111
$973$	−54600.2	−1.79898
$974$	27666.1	0.910144
$975$	10039.9	0.329779
$976$	17722.7	0.581241
$977$	40922.8	1.34006	0.670030	−	0.742334i	$-0.266281\pi$
$977$	40922.8	1.34006	0.670030	+	0.742334i	$0.266281\pi$
$978$	20039.7	0.655215
$979$	11002.0	0.359169
$980$	4824.73	0.157265
$981$	−6178.28	−0.201078
$982$	34327.1	1.11550
$983$	−17232.4	−0.559133	−0.279566	−	0.960126i	$-0.590191\pi$
$983$	−17232.4	−0.559133	−0.279566	+	0.960126i	$0.590191\pi$
$984$	−51333.7	−1.66307
$985$	−9124.42	−0.295155
$986$	12876.6	0.415896
$987$	2254.95	0.0727212
$988$	2893.98	0.0931880
$989$	−33457.8	−1.07573
$990$	−1896.18	−0.0608733
$991$	−12670.6	−0.406150	−0.203075	−	0.979163i	$-0.565094\pi$
$991$	−12670.6	−0.406150	−0.203075	+	0.979163i	$0.565094\pi$
$992$	22777.9	0.729032
$993$	24211.0	0.773728
$994$	−39493.0	−1.26020
$995$	−26547.2	−0.845831
$996$	4773.83	0.151872
$997$	2673.09	0.0849123	0.0424562	−	0.999098i	$-0.486482\pi$
$997$	2673.09	0.0849123	0.0424562	+	0.999098i	$0.486482\pi$
$998$	11997.5	0.380536
$999$	−15349.9	−0.486137

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.c.1.6	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.c.1.6	✓	20	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1045.a (trivial)

\(f(q)\)	\(=\)	\(q-2.35645 q^{2} -6.45217 q^{3} -2.44713 q^{4} -5.00000 q^{5} +15.2042 q^{6} +27.1536 q^{7} +24.6182 q^{8} +14.6305 q^{9} +O(q^{10})\)	\(q-2.35645 q^{2} -6.45217 q^{3} -2.44713 q^{4} -5.00000 q^{5} +15.2042 q^{6} +27.1536 q^{7} +24.6182 q^{8} +14.6305 q^{9} +11.7823 q^{10} -11.0000 q^{11} +15.7893 q^{12} -62.2421 q^{13} -63.9861 q^{14} +32.2608 q^{15} -38.4345 q^{16} -24.6401 q^{17} -34.4760 q^{18} +19.0000 q^{19} +12.2357 q^{20} -175.199 q^{21} +25.9210 q^{22} -87.4691 q^{23} -158.841 q^{24} +25.0000 q^{25} +146.670 q^{26} +79.8102 q^{27} -66.4484 q^{28} +221.769 q^{29} -76.0211 q^{30} -214.126 q^{31} -106.376 q^{32} +70.9739 q^{33} +58.0631 q^{34} -135.768 q^{35} -35.8027 q^{36} -192.330 q^{37} -44.7726 q^{38} +401.596 q^{39} -123.091 q^{40} +323.178 q^{41} +412.849 q^{42} +382.510 q^{43} +26.9185 q^{44} -73.1524 q^{45} +206.117 q^{46} -12.8708 q^{47} +247.986 q^{48} +394.316 q^{49} -58.9113 q^{50} +158.982 q^{51} +152.315 q^{52} -345.779 q^{53} -188.069 q^{54} +55.0000 q^{55} +668.471 q^{56} -122.591 q^{57} -522.587 q^{58} +318.805 q^{59} -78.9466 q^{60} -461.116 q^{61} +504.577 q^{62} +397.270 q^{63} +558.147 q^{64} +311.210 q^{65} -167.246 q^{66} +868.796 q^{67} +60.2975 q^{68} +564.365 q^{69} +319.930 q^{70} +617.212 q^{71} +360.176 q^{72} +1124.99 q^{73} +453.217 q^{74} -161.304 q^{75} -46.4955 q^{76} -298.689 q^{77} -946.342 q^{78} +611.159 q^{79} +192.172 q^{80} -909.972 q^{81} -761.552 q^{82} +302.346 q^{83} +428.736 q^{84} +123.200 q^{85} -901.366 q^{86} -1430.89 q^{87} -270.800 q^{88} -1000.18 q^{89} +172.380 q^{90} -1690.09 q^{91} +214.049 q^{92} +1381.57 q^{93} +30.3293 q^{94} -95.0000 q^{95} +686.358 q^{96} -599.833 q^{97} -929.188 q^{98} -160.935 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9}+O(q^{10}) \) 20 * q - q^2 - 8 * q^3 + 51 * q^4 - 100 * q^5 - 54 * q^6 + 49 * q^7 + 9 * q^8 + 146 * q^9	\( 20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9} + 5 q^{10} - 220 q^{11} - 59 q^{12} + 60 q^{13} - 89 q^{14} + 40 q^{15} + 275 q^{16} - 155 q^{17} + 45 q^{18} + 380 q^{19} - 255 q^{20} + 105 q^{21} + 11 q^{22} - 154 q^{23} - 397 q^{24} + 500 q^{25} + 176 q^{26} - 206 q^{27} + 155 q^{28} - 305 q^{29} + 270 q^{30} - 759 q^{31} - 254 q^{32} + 88 q^{33} - 565 q^{34} - 245 q^{35} + 705 q^{36} + 698 q^{37} - 19 q^{38} - 758 q^{39} - 45 q^{40} + 547 q^{41} + 106 q^{42} - 925 q^{43} - 561 q^{44} - 730 q^{45} - 254 q^{46} - 681 q^{47} - 540 q^{48} + 213 q^{49} - 25 q^{50} - 899 q^{51} + 889 q^{52} - 419 q^{53} - 2241 q^{54} + 1100 q^{55} - 2473 q^{56} - 152 q^{57} - 1440 q^{58} - 2829 q^{59} + 295 q^{60} - 959 q^{61} + 1575 q^{62} - 426 q^{63} + 93 q^{64} - 300 q^{65} + 594 q^{66} - 1020 q^{67} - 4218 q^{68} - 572 q^{69} + 445 q^{70} + 106 q^{71} + 210 q^{72} + 558 q^{73} - 3439 q^{74} - 200 q^{75} + 969 q^{76} - 539 q^{77} - 3599 q^{78} + 536 q^{79} - 1375 q^{80} - 2128 q^{81} - 1255 q^{82} - 4179 q^{83} - 2024 q^{84} + 775 q^{85} - 1119 q^{86} - 557 q^{87} - 99 q^{88} - 4120 q^{89} - 225 q^{90} - 111 q^{91} - 2831 q^{92} + 801 q^{93} + 1213 q^{94} - 1900 q^{95} - 6147 q^{96} + 1414 q^{97} - 7869 q^{98} - 1606 q^{99}+O(q^{100}) \) 20 * q - q^2 - 8 * q^3 + 51 * q^4 - 100 * q^5 - 54 * q^6 + 49 * q^7 + 9 * q^8 + 146 * q^9 + 5 * q^10 - 220 * q^11 - 59 * q^12 + 60 * q^13 - 89 * q^14 + 40 * q^15 + 275 * q^16 - 155 * q^17 + 45 * q^18 + 380 * q^19 - 255 * q^20 + 105 * q^21 + 11 * q^22 - 154 * q^23 - 397 * q^24 + 500 * q^25 + 176 * q^26 - 206 * q^27 + 155 * q^28 - 305 * q^29 + 270 * q^30 - 759 * q^31 - 254 * q^32 + 88 * q^33 - 565 * q^34 - 245 * q^35 + 705 * q^36 + 698 * q^37 - 19 * q^38 - 758 * q^39 - 45 * q^40 + 547 * q^41 + 106 * q^42 - 925 * q^43 - 561 * q^44 - 730 * q^45 - 254 * q^46 - 681 * q^47 - 540 * q^48 + 213 * q^49 - 25 * q^50 - 899 * q^51 + 889 * q^52 - 419 * q^53 - 2241 * q^54 + 1100 * q^55 - 2473 * q^56 - 152 * q^57 - 1440 * q^58 - 2829 * q^59 + 295 * q^60 - 959 * q^61 + 1575 * q^62 - 426 * q^63 + 93 * q^64 - 300 * q^65 + 594 * q^66 - 1020 * q^67 - 4218 * q^68 - 572 * q^69 + 445 * q^70 + 106 * q^71 + 210 * q^72 + 558 * q^73 - 3439 * q^74 - 200 * q^75 + 969 * q^76 - 539 * q^77 - 3599 * q^78 + 536 * q^79 - 1375 * q^80 - 2128 * q^81 - 1255 * q^82 - 4179 * q^83 - 2024 * q^84 + 775 * q^85 - 1119 * q^86 - 557 * q^87 - 99 * q^88 - 4120 * q^89 - 225 * q^90 - 111 * q^91 - 2831 * q^92 + 801 * q^93 + 1213 * q^94 - 1900 * q^95 - 6147 * q^96 + 1414 * q^97 - 7869 * q^98 - 1606 * q^99

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